Multicontinuum homogenization in perforated domains

Abstract

In this paper, we develop a general framework for multicontinuum homogenization in perforated domains. The simulations of problems in perforated domains are expensive and, in many applications, coarse-grid macroscopic models are developed. Many previous approaches include homogenization, multiscale finite element methods, and so on. In our paper, we propose multicontinuum homogenization method for problems in perforated domains. We distinguish different spatial regions in perforations based on their sizes. For example, very thin perforations are considered as one continuum, while larger perforations are considered as another continuum. By differentiating perforations in this way, we are able to predict flows in each of them more accurately. Cell problems are formulated for each continuum using appropriate constraints for the solution averages and their gradients. These cell problem solutions are used in a multiscale expansion and in deriving novel macroscopic systems for multicontinuum homogenization. Our proposed approaches are designed for problems without scale separation. We present numerical results for two continuum problems and demonstrate the accuracy of the proposed methods.